Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 11^{2} \cdot 167 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 2·5-s + 4·7-s − 8-s + 2·10-s − 4·14-s + 16-s − 4·17-s + 4·19-s − 2·20-s + 4·23-s − 25-s + 4·28-s − 2·29-s + 4·31-s − 32-s + 4·34-s − 8·35-s − 12·37-s − 4·38-s + 2·40-s + 12·41-s + 8·43-s − 4·46-s + 9·49-s + 50-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.894·5-s + 1.51·7-s − 0.353·8-s + 0.632·10-s − 1.06·14-s + 1/4·16-s − 0.970·17-s + 0.917·19-s − 0.447·20-s + 0.834·23-s − 1/5·25-s + 0.755·28-s − 0.371·29-s + 0.718·31-s − 0.176·32-s + 0.685·34-s − 1.35·35-s − 1.97·37-s − 0.648·38-s + 0.316·40-s + 1.87·41-s + 1.21·43-s − 0.589·46-s + 9/7·49-s + 0.141·50-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 363726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 363726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(363726\)    =    \(2 \cdot 3^{2} \cdot 11^{2} \cdot 167\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{363726} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 363726,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.990691272$
$L(\frac12)$  $\approx$  $1.990691272$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;11,\;167\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;11,\;167\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 \)
11 \( 1 \)
167 \( 1 - T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−12.15758079572043, −12.06705829476458, −11.48542929344832, −11.07940407294798, −10.83451380170991, −10.48371384302341, −9.634360996803020, −9.253787972338776, −8.853570298591095, −8.351775214364208, −7.921501564603488, −7.571360256904557, −7.226128034341317, −6.703922447735441, −5.998321003778611, −5.534733565705241, −4.876257981112690, −4.533328645175973, −4.081264135643909, −3.305942510078375, −2.921731671956391, −2.028242630069764, −1.773782257251722, −0.9550627359723373, −0.4874604149085451, 0.4874604149085451, 0.9550627359723373, 1.773782257251722, 2.028242630069764, 2.921731671956391, 3.305942510078375, 4.081264135643909, 4.533328645175973, 4.876257981112690, 5.534733565705241, 5.998321003778611, 6.703922447735441, 7.226128034341317, 7.571360256904557, 7.921501564603488, 8.351775214364208, 8.853570298591095, 9.253787972338776, 9.634360996803020, 10.48371384302341, 10.83451380170991, 11.07940407294798, 11.48542929344832, 12.06705829476458, 12.15758079572043

Graph of the $Z$-function along the critical line